I have some question about the last conclusion in the proof of Lemma 2.16 on page 20 of the paper "Hall's theorem for limit groups". You can find the paper on

https://arxiv.org/abs/math/0605546

So. After assuming that for infinitely many indices $p$, $q_p$ and some $b_k$ we have that

$$z^p[b_k,z^{q_p}]z^{-p}\in\pi_1(Y')$$

the author concludes that $[b_k,z]=1$, since the elevations $\partial_{1,2}^+$ are proper. If i am right, he applies the previous Lemma, Lemma 2.15 on page 19. But to do so, i think, one has to assume that there exists some fixed $g\in G$ such that for infinitely many indices $p$ the element $g_p=[b_k,z^{q_p}]$ equals $g$. But how is that possible? The element $g_p$ depend on $p$. So couldn't it happen that for any $p,p'$ we have $g_p\neq g_{p'}$?

Thanks for your response!

PS: I'm sorry, if this is rather a question for mathstackexchange.

I have some question about the last conclusion in the proof of Lemma 2.16 on page 20 of the paper "Hall's theorem for limit groups". You can find the paper on

https://arxiv.org/abs/math/0605546

So. After assuming that for infinitely many indices $p$, $q_p$ and some $b_k$ we have that

$$z^p[b_k,z^{q_p}]z^{-p}\in\pi_1(Y')$$

the author concludes that $[b_k,z]=1$, since the elevations $\partial_{1,2}^+$ are proper. If i am right, he applies the previous Lemma, Lemma 2.15 on page 19. But to do so, i think, one has to assume that there exists some fixed $g\in G$ such that for infinitely many indices $p$ the element $g_p=[b_k,z^{q_p}]$ equals $g$. But how is that possible? The element $g_p$ depends on $p$. So couldn't it happen that for any $p,p'$ with $p\neq p'$ we have $g_p\neq g_{p'}$?

Thanks for your response!

PS: I'm sorry, if this is rather a question for mathstackexchange.